Acronym ...
Name 2quiphi (?)
Circumradius sqrt[5-2 sqrt(5)] = 0.726543
General of army rox
Colonel of regiment rigfix
Confer
non-Grünbaumian master:
quiphi  

Looks like a compound of 2 quasiprismatohecatonicosachora (quiphi), and indeed all elements coincide by pairs.

As abstract polytope quiphi is isomorphic to sophi, thereby replacing pentagrams by pentagons, resp. by replacing stip by pip and sissid by gad.


Incidence matrix according to Dynkin symbol

x5/2o5/4o5/2x

.   .   .   . | 1440 |    5    5 |    5   10    5 |   1   5   5   1
--------------+------+-----------+----------------+----------------
x   .   .   . |    2 | 3600    * |    2    2    0 |   1   2   1   0
.   .   .   x |    2 |    * 3600 |    0    2    2 |   0   1   2   1
--------------+------+-----------+----------------+----------------
x5/2o   .   . |    5 |    5    0 | 1440    *    * |   1   1   0   0
x   .   .   x |    4 |    2    2 |    * 3600    * |   0   1   1   0
.   .   o5/2x |    5 |    0    5 |    *    * 1440 |   0   0   1   1
--------------+------+-----------+----------------+----------------
x5/2o5/4o   .    12 |   30    0 |   12    0    0 | 120   *   *   *
x5/2o   .   x    10 |   10    5 |    2    5    0 |   * 720   *   *
x   .   o5/2x    10 |    5   10 |    0    5    2 |   *   * 720   *
.   o5/4o5/2x    12 |    0   30 |    0    0   12 |   *   *   * 120
or
.   .   .   .    | 1440 |   10 |   10   10 |   2   10
-----------------+------+------+-----------+---------
x   .   .   .  & |    2 | 7200 |    2    2 |   1    3
-----------------+------+------+-----------+---------
x5/2o   .   .  & |    5 |    5 | 2880    * |   1    1
x   .   .   x    |    4 |    4 |    * 3600 |   0    2
-----------------+------+------+-----------+---------
x5/2o5/4o   .  &    12 |   30 |   12    0 | 240    *
x5/2o   .   x  &    10 |   15 |    2    5 |   * 1440

x5/2o5o5/3x

.   . .   . | 1440 |    5    5 |    5   10    5 |   1   5   5   1
------------+------+-----------+----------------+----------------
x   . .   . |    2 | 3600    * |    2    2    0 |   1   2   1   0
.   . .   x |    2 |    * 3600 |    0    2    2 |   0   1   2   1
------------+------+-----------+----------------+----------------
x5/2o .   . |    5 |    5    0 | 1440    *    * |   1   1   0   0
x   . .   x |    4 |    2    2 |    * 3600    * |   0   1   1   0
.   . o5/3x |    5 |    0    5 |    *    * 1440 |   0   0   1   1
------------+------+-----------+----------------+----------------
x5/2o5o   .    12 |   30    0 |   12    0    0 | 120   *   *   *
x5/2o .   x    10 |   10    5 |    2    5    0 |   * 720   *   *
x   . o5/3x    10 |    5   10 |    0    5    2 |   *   * 720   *
.   o5o5/3x    12 |    0   30 |    0    0   12 |   *   *   * 120

x5/3o5/4o5/3x

.   .   .   . | 1440 |    5    5 |    5   10    5 |   1   5   5   1
--------------+------+-----------+----------------+----------------
x   .   .   . |    2 | 3600    * |    2    2    0 |   1   2   1   0
.   .   .   x |    2 |    * 3600 |    0    2    2 |   0   1   2   1
--------------+------+-----------+----------------+----------------
x5/3o   .   . |    5 |    5    0 | 1440    *    * |   1   1   0   0
x   .   .   x |    4 |    2    2 |    * 3600    * |   0   1   1   0
.   .   o5/3x |    5 |    0    5 |    *    * 1440 |   0   0   1   1
--------------+------+-----------+----------------+----------------
x5/3o5/4o   .    12 |   30    0 |   12    0    0 | 120   *   *   *
x5/3o   .   x    10 |   10    5 |    2    5    0 |   * 720   *   *
x   .   o5/3x    10 |    5   10 |    0    5    2 |   *   * 720   *
.   o5/4o5/3x    12 |    0   30 |    0    0   12 |   *   *   * 120
or
.   .   .   .    | 1440 |   10 |   10   10 |   2   10
-----------------+------+------+-----------+---------
x   .   .   .  & |    2 | 7200 |    2    2 |   1    3
-----------------+------+------+-----------+---------
x5/3o   .   .  & |    5 |    5 | 2880    * |   1    1
x   .   .   x    |    4 |    4 |    * 3600 |   0    2
-----------------+------+------+-----------+---------
x5/3o5/4o   .  &    12 |   30 |   12    0 | 240    *
x5/3o   .   x  &    10 |   15 |    2    5 |   * 1440

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